This notion is Popper's, and it is a good one, but is not universally held.
During periods of normal science, or when total falsification can be justified without depending deeply on a theory, falsification gives the most statistically stable means of moving forward.
The entire mechanism of modern statistics is built around the 'null hypothesis', and stating up front exactly what it would mean for a theory to fail to account for experimental results. This boils 'partial failure' down into measurable numbers and allows us to accumulated them and trust what they mean in terms of our real likelihood of being wrong.
So when you can use this criterion and apply these methods, you can converge on the most reliable answer in a way that can be validated by one's peers, and can give them some notion of how much risk is involved in accepting the theory if it is marginal.
Of course, where there are people involved, the math is not completely trustworthy, because practice involves interpretation, etc., and scientists still have judgment calls to make. But when Popper applies, it is gives decisions a lot of accountability.
But it remains an open question whether all science should be approached in this way, even when it can. Many folks after Popper have contested the power of this method when things are less stable. Kuhn, for example has suggested that science regularly moves into 'revolutionary' periods, when determining falsifiability is not always possible, and therefore it cannot be required. (Feyerabend goes even further, claiming that science is constantly at least partially in such a state, and should attempt to remain so.)
When a science is trading out older paradigms whose creative energy has been spent, and deriving new paradigms, the old and the new can be incommensurable in a way that obscures our ability to know what exactly 'falsified' means in the context of the science in flux.
When a science is close to a revolutionary boundary, just descending into or just getting out of a paradigm switch, demanding falsifiability it is not always the best choice. It may shoot down new ideas based on habits and older ideas that are about to be shot down, themselves. Potential new paradigms need slack to establish a footing, and preserving parts of the old paradigm that still happen to remain clear in the new interpretation may undermine cohesiveness in the new foundation coming into practice, which will make for a complicated mess in the future.
If, like Feyerabend, we prefer every science should always have competing sets of underlying assumptions, and avoid 'hegemonizing', then falsifiability becomes something optional, if still very convenient, in a theory.
You can easily claim that the main parts of psychology are not supported by a single underlying paradigm, but that psychology is so young a science that it is continually in revolution, and will continue to have competing basic paradigms for the foreseeable future.
That excuses the fact that its basic motivating theories, from psychoanalysis to Cognitive Behaviorism are basically non-falsifiable, but should not simply be dismissed as non-science.
And you can see how good statistical practice is still useful, even though there is no ultimate chance of falsifying, e.g. the idea of the ego.